Theoretical Light Curves of Type Ia Supernovae

A&A 528, A141 (2011) Astronomy DOI: 10.1051/0004-6361/201014778 & c ESO 2011 Astrophysics

Theoretical light curves of type Ia supernovae

D. Jack1, P. H. Hauschildt1,andE.Baron1,2

1 Hamburger Sternwarte, Gojenbergsweg 112, 21029 Hamburg, Germany e-mail: [djack;yeti]@hs.uni-hamburg.de 2 Homer L. Dodge Department of Physics and Astronomy, University of Oklahoma, 440 W Brooks, Rm 100, Norman, OK 73019-2061, USA e-mail: [email protected] Received 13 April 2010 / Accepted 9 February 2011

ABSTRACT

Aims. We present the first theoretical SN Ia light curves calculated with the time-dependent version of the general purpose model atmosphere code PHOENIX. Our goal is to produce light curves and spectra of hydro models of all types of supernovae. Methods. We extend our model atmosphere code PHOENIX to calculate type Ia supernovae light curves. A simple solver was implemented which keeps track of energy conservation in the atmosphere during the free expansion phase. Results. The correct operation of the new additions to PHOENIX were verified in test calculations. Furthermore, we calculated theoretical light curves and compared them to the observed SN Ia light curves of SN 1999ee and SN 2002bo. We obtained LTE as well as NLTE model light curves. Conclusions. We have verified the correct operation of our extension into the time domain. We have calculated the first SN Ia model light curves using PHOENIX in both LTE and NLTE. For future work the infrared model light curves need to be further investigated. Key words. supernovae: general – radiative transfer – methods: numerical

1. Introduction light curves of type Ia supernova events. In a further section, we present results of NLTE model light curve calculations. All types of supernovae are important for the role that they play in understanding stellar evolution, galactic nucleosynthesis, and as cosmological probes. Type Ia supernovae are of particular 2. Energy solver cosmological interest, e.g., because the dark energy was discov- In our previous paper (Jack et al. 2009), we presented our ap- ered with type Ia supernovae (Riess et al. 1998; Perlmutter et al. proach to calculate light curves of type Ia supernovae events. We 1999). implemented a simple energy solver into our general purpose In dark energy studies, the goal now is to characterize the model atmosphere code PHOENIX where we kept track of the nature of the dark energy as a function of redshift. While there overall energy change of the radiating fluid and the energy ex- are other probes that will be used (gravitational lensing, baryon change between the matter and radiation. Simple test light curves acoustic oscillations), a JDEM or Euclid mission will likely con- confirmed that our approach worked correctly. We now present sider supernovae in some form. In planning for future dark en- a new approach to calculate theoretical light curves of SNe Ia. ergy studies both from space and from the ground, it is important Again, we are using a simple solver where we keep track of the to know whether the mission will require spectroscopy of mod- energy conservation. In the new approach, we consider only the est resolution, or whether pure imaging or grism spectroscopy energy density of the material. We assume free expansion and do will be adequate. Several purely spectral indicators of peak lumi- not solve the equations of hydrodynamics. In order to obtain the nosity have been proposed (Nugent et al. 1995; Hachinger et al. result for the new time step, we use an explicit scheme. We apply 2006; Bongard et al. 2006; Bronder et al. 2008; Foley et al. 2008; this scheme and compute time steps until radiative equilibrium Chotard et al. 2011). What is required is an empirical and theo- is reached. These are the points in our light curves. retical comparison of both light curve shape luminosity indica- The direct change of the energy density of the material con- tors (Pskovskii 1977; Phillips 1993; Riess et al. 1996; Goldhaber sidering absorption and emission of radiation and energy deposi- et al. 2001) and spectral indicators. tion by gamma rays is given by Eq. (96.7) in Mihalas & Mihalas To make this comparison one needs to know more about the (1984) physics going on in a supernova explosion and to be able to cal- de d 1 culate light curves and spectra self-consistently. Thus, we need ρ + = χ − πη + ρ, p ρ (c E 4 ) (1) to extend our code to time-dependent problems. While our pri- dt dt mary focus is on type Ia supernovae, the time-dependent radia- where ρ is the density, p the gas pressure and e is the energy tive transfer code is applicable to all types of supernovae, as well density of the material. The quantities of the radiation field are as to other objects, e.g., stellar pulsations. χ which is the absorption coefficient, η is the emission coeffi- = 4π In the following we present the methods we used to imple- cient and E c J is the radiation energy density with the mean ment time dependence. First we focus on solving the energy intensity J. To obtain E, the radiative transfer equation in spheri- equation (first law) and then present our first theoretical LTE cal symmetry is solved including special relativity. All additional Article published by EDP Sciences A141, page 1 of 11 A&A 528, A141 (2011) energy sources are put in such as the energy input from gamma During the first phase of the SN Ia envelope evolution, the ray deposition. This equation represents the first law of thermo- material of the atmosphere is hot and, therefore, highly ionized. dynamics for the material. The change of the energy density of The energy change due to ionization and excitation changes of the material depends on the coupling of matter and radiation the atoms present in the SN Ia atmosphere cannot be neglected. field, the absorption of gamma-radiation and positron annihila- PHOENIX already solves the equation of state (EOS), where all tion energy, the expansion work and the change of the ionization the excitation and ionization stages of the present atoms and and excitation of matter. molecules are included. Using the EOS, we obtain the overall Dividing equation 1 by the material density ρ we obtain energy density of the material by the sum of the ionization energy eions and the translational energy etrans de 1 d 1 = (cχE − 4πη) dλ − p + . (2) dt ρ dt ρ e = etrans + eions. (8) = 4π Hence, the energy density change of the material goes into a The radiation energy density is given by E c J,whereJ is the mean intensity. Using this, we obtain for the radiation term change of the translational energy and the ionization energy, which both depend on the temperature. Therefore, we obtain the new temperature by an iteration scheme. The matter density at (cχE − 4πη)dλ = 4π (χJ − η)dλ = 4π (χ(J − S ))dλ, (3) the next point in time is determined by homologous expansion. A first temperature guess is used, and the EOS is solved to obtain = η where S χ is the source function. All these quantities can the ionization energy density. Combined with the translational be derived from the solution of the radiative transfer equation. energy density, the overall energy density is computed. This is The term of the change of the energy density by the radiation is checked against the target energy density, which we obtained therefore given by Eq. (6). If the obtained energy density is incorrect, a new temperature guess is made. This new temperature guess is obtained Q = χλ(Jλ − S λ)dλ. (4) by assuming a linear dependence of the energy density and temperature. The current temperature guess is iterated to the target energy density. It takes about 5–10 iteration steps to determine Another change of the energy density is due to the work W done the new temperature. If the EOS delivers the correct target en- by the adiabatic expansion of the SN Ia atmosphere. For a dis- ergy density, the new temperature of the next time step has been crete step this work is given by found. The accuracy relative of the energy density in this itera- −5 1 1 tion process is set to 10 . W = p − · (5) ρ2 ρ1 2.1. γ-ray deposition The expansion is assumed to be homologous. Since we solve the energy equation for the matter, we do not include the radiation The maximum of the light curve of an SN Ia event is observed pressure work. Since the system is radiation dominated there is around 20 days after explosion. Causing this later maximum of the possibility of numerical inaccuracies in coupling the matter the light curve of an SN Ia event is the energy release into the and radiation only by Q. For the calculation of the new radii envelope caused by the radioactive decay of 56Ni and its also ra- and densities as well as a discussion about the accuracy of this dioactive decay product 56Co. Therefore, this energy deposition assumption see Jack et al. (2009). has a strong influence on the energy change of the SN Ia enve- Including all energy changing effects, the new energy den- lope structure. Hence, the energy deposition due to radioactive sity of the material e2 after a discrete time step Δt is explicitly decay has to be taken into account for the calculation of the SN given by Ia envelope evolution. The energy deposition due to the γ-rays emitted by radioac- π = − 1 − 1 + 4 Δ χ − λ + Δ , tive isotopes needs to be computed by a radiative transfer solver e2 e1 p t (J S )d t (6) γ γ ρ2 ρ1 ρ for the -rays. In this work, we solve the -ray deposition with the assumption of a gray atmosphere for the γ-rays. Jeffery where e1 is the old energy density. (1998) did a detailed study of the γ-ray deposition and pointed To obtain all the needed quantities we have to solve the out that this is an adequate approach to calculate γ-ray depo- spherically symmetric special relativistic radiative transfer equa- sition in SN Ia atmospheres. In the decay of a 56Ni nucleus, a tion. The advantage of our new approach is that we do not have γ-photon is emitted with an energy of 2.136 MeV. The 56Co nu- to iterate for each time step. In our previous paper the calcula- cleus decays to an 56Fe nucleus and emits a γ-photon, which tion of each time step involved an iteration process to obtain the has an energy of 4.566 MeV. In the decay of 56Co about 19% of new matter temperature. With the new approach we can calcu- the energy is released by positrons. The positrons are assumed to late the new temperature for the next time step directly from all be locally trapped. They annihilate by emitting two photons each the known quantities. with an energy of 512 keV,which has to be taken into account for The translational energy density of the material is given by the energy deposition calculation. The opacity is considered to be constant and a pure absorption opacity, meaning that no scat- 3 p 3 R 3 NAkT ff κ = . / 2 −1 etrans = = T = , (7) tering is assumed. As in Je ery (1998), γ 0 06 Z A cm g 2 ρ 2 μ 2 m was chosen as the opacity. Z/A is the proton fraction, which with the mean molecular weigh μ and the universal gas constant is counts both bound and free electrons, since the electron bind- R. The gas pressure is represented by p and the density by ρ. T ing energy is small compared to the energy of gamma rays. The energy deposition into the atmosphere per unit time is given by stands for the temperature of the material. This equation of the χ energy density is now used to determine the new temperature = π , after the next time step. 4 ρ J (9)

A141, page 2 of 11 D. Jack et al.: Theoretical light curves of type Ia supernovae where J is the mean intensity, which has been obtained by solving the gray radiative transfer for the γ-rays. This obtained energy deposition has to be taken into account for the calculation of the overall energy change.

2.2. Adaptive time step procedure The timescale for energy changes in the SN Ia envelope will change during the evolution of the light curve. In order to save computation time, the light curves have to be calculated with the optimal time step size for each phase of light curve evolution. Therefore, we implemented an adaptive time step routine to determine the optimal time step size for the current time step. The energy change Δe of the energy of the material e may be approximated by Δe = x · e =Δt · (Q + ) , (10) Fig. 1. Temperature structures of a test atmosphere obtained with the where Q is the energy change of the interaction with the radia- energy solver and the PHOENIX temperature correction procedure. tion, and is the energy deposition by the gamma rays. The energy change due to the expansion is ignored in this case. On the one hand, this energy change depends on the new matter density temperature structure of an SN Ia precisely. However, for the after the time step, which is unknown because it depends on the test calculations for our energy solver, this simple temperature structure is sufficient. With this initial atmosphere structure, the size of the time step itself. Furthermore, the energy change be- ff cause of the expansion is small compared to the changes caused energy solver is applied for di erent test cases. All contributions by the energy transport and the energy deposition by γ-rays. The to the energy change are tested separately. idea of the adaptive time step procedure is to limit the energy change to a prescribed amount of the energy of the material. 3.1. Energy transport Thus, rewriting Eq. (10), we obtain the time step size Δt for the In this section, the energy transport through the atmosphere current time step by is tested. The energy solver considers only the energy change caused by emission and absorption of radiation, where the result Δ = e · , t + x (11) of the radiative transfer equation is needed. All other influences Q are neglected. As a first test, we check how the initial tempera- where x is the introduced limiting energy change factor. The fac- ture structure changes if the energy solver is changing the SN Ia tor x ranges between xmin and xmax, which mark the largest and envelope. As the initial atmosphere structure is already in radia- smallest allowed energy change. These are input parameters for tive equilibrium, the energy solver should not change the tem- the adaptive time step procedure. The time step size is calculated perature structure significantly, because it also pushes the atmo- for every layer, and the minimum time step size of all layers is sphere towards a radiative equilibrium state. used for the energy solver. In Fig. 1, a comparison of the temperature structure of the Each time the adaptive time step procedure is called, it energy solver to the result of the temperature correction proce- checks if the energy changes of the time step before were too dure is shown. The differences in the temperature structure for big or could have been bigger. If the minimum time step size is most layers are less than 1%. But the temperature differences of in a different layer, the same value of x is kept for the next time the inner layers are clearly higher. These differences arise in the step. If the minimum is in the same layer and the sign of the en- PHOENIX temperature correction result which produces a spike ergy change does not change, the previous time step might have in the temperature structure. This is likely due to the bound- been too small. Thus, the factor x is increased for the following ary condition in the temperature correction. Hence, the resulting time step. If the sign changes, the last time step might have been temperature structure obtained with the energy solver is more too large, therefore, the factor x is decreased. This means that accurate. Here, the temperature structure is smooth. In order to for each time step, the allowed energy change is adapted and the obtain an atmosphere in radiative equilibrium, the energy trans- factor x is updated to get the optimal time step during the whole port part of the energy solver can be used instead of the tempera- evolution of the SN Ia atmosphere. ture correction procedure. The main problem is that about a few hundred time steps are needed to obtain the resulting atmosphere structure in radiative equilibrium, while the temperature correc- 3. Test calculations tion needs fewer iteration steps and is, therefore, significantly All new implemented physical processes of the simple energy faster. solver have to be tested. For the test atmosphere, the atmosphere structure and abundances of the W7 deflagration model (Nomoto 3.2. Expansion 1984) are used. The atmosphere structure is expanded to a point In a further test calculation, the expansion part of the energy in time of 10 days after the explosion. The densities and radii are solver is checked. The only energy change considered is the adi- determined by free homologous expansion and can be computed abatic cooling due to free expansion of the SN Ia envelope. The easily. To perform the test calculations, we obtained an initial energy deposition by γ-rays or an energy change due to energy temperature structure with the PHOENIX temperature correction transport is disabled. For this test case, the expectation is that procedure (Hauschildt et al. 2003). This temperature structure the atmosphere should just cool down, so the temperature of the is the result of a simple approach that does not represent the atmosphere and the observed luminosity should be decreasing.

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Fig. 2. Light curve of a test atmosphere that is just expanding and adia- Fig. 4. Light curve of a test atmosphere with energy deposition. Because batically cooling. of the additional energy, the luminosity increases with time.

Fig. 3. Initial and final temperature structure of a test atmosphere which Fig. 5. Model light curve of a realistic test scenario. is just expanding and adiabatically cooling. expansion as well as energy deposition and energy transport are The observed luminosity is shown in Fig. 2. The observed active for this computation. luminosity of the SN Ia atmosphere decreases in time. The tem- The observed light curve is shown in Fig. 5. The luminosity perature structure of the first and the last time step is plotted in increases because of the energy input from deposition of energy Fig. 3. The adiabatic expansion has cooled the atmosphere ev- from the γ-rays produced by decay of nickel and cobalt. It takes erywhere, and the new temperature structure is now significantly a certain time, until the whole atmosphere has relaxed to this lower than the initial one. new condition. The atmosphere is then in radiative equilibrium state, and the luminosity stays constant. The initial and final tem- 3.3. Energy deposition perature structure are plotted in Fig. 6. The energy input caused To test the energy deposition by the radioactive decay of nickel by radioactive decay has increased the temperature of the whole and cobalt into the SN Ia atmosphere, a test case is considered, atmosphere. The atmosphere is heated by γ-ray deposition in where only this γ-ray deposition is calculated with the energy the inner part of the atmosphere. Due to this additional energy, solver. The energy change due to free expansion and energy the luminosity of these layers increases and the heat is radiated transport are neglected to see the direct effect of the additional away and absorbed by the surrounding layers. This energy trans- energy put into the test SN Ia atmosphere. port takes care that the deposited energy is moving through the The results of the energy deposition by γ-rays calculation whole atmosphere so that the temperature increases everywhere with the energy solver is shown in Fig. 4, where the light curve and the additional energy from the radioactive decay is radiated is plotted. Due to the energy added to the atmosphere, the lumi- away towards the observer. The atmosphere is then in radiative nosity seen by an observer increases with time. equilibrium.

3.4. Realistic test scenario 4. SN Ia model light curves After all single effects have been tested, we now calculate a test case where all effects are considered for the solution of the en- We now present our first theoretical SN Ia light curves ob- ergy solver. Again, the same initial temperature structure is used tained with our extensions to the general purpose model atmo- and we start our calculation at day 10 after the explosion. Free sphere code PHOENIX and compare them to observed SN Ia light

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Fig. 6. The initial temperature structure for the static solution of the Fig. 7. A typical model spectrum of an SN Ia together with the filter equation of radiative equilibrium and the relaxed temperature structure functions for the U, B, V, R and I band. These are used to determine the obtained 1 min later by advancing the energy solver in the test scenario. luminosity in each band for the model light curves. All influences on the SN Ia atmosphere are considered. equilibrium state. A typical time step of an energy change in the curves. The online supernova spectrum archive (SUSPECT) model atmosphere is about 0.1 s. It takes about 500 time steps (Richardson et al. 2001, 2002)1 provides numerous of obser- ff to reach radiative equilibrium depending on in which evolution vations of di erent types of supernovae. For this work, the ob- phase the SN Ia is. For the later phase after the maximum of the served light curves of SN 2002bo and SN 1999ee are used to SN Ia light curve, fewer time steps are needed. We use this first compare them to our results of model light curves. Photometric ff radiative equilibrium atmosphere structure to calculate a more light curve observations of SN 2002bo in di erent photomet- detailed spectrum, where more wavelength points are used. This ric bands (Benetti et al. 2004) have been obtained. SN 1999ee model spectrum is then used to obtain the first point of the model also has observed spectra (Hamuy et al. 2002) and photometry light curve for each band by using the filter functions described (Stritzinger et al. 2002). below. It certainly would require too much computation time to calculate a whole light curve evolution by using the typical 4.1. Method time steps of about 0.1 s. Therefore, to obtain the next point of The energy solver is now applied to calculate synthetic light the light curve, big time steps are computed. For these big time curves of SNe Ia. The SN Ia light curve evolution is calculated steps, the atmosphere is only expanding. This means that nei- during the free expansion phase. For the initial model atmo- ther energy deposition by γ-rays or energy transport through the sphere structure, the results of the explosion calculation of other atmosphere is considered for the solution of the energy solver. groups are used as the input structure. Each layer has a certain After half a day computed with big time steps, the whole en- expansion velocity, which does not change during the evolution, ergy solver changes the atmosphere structure again and the next because homologous expansion is assumed. We start the model point in the light curve is obtained after the atmosphere structure light curve calculation a few days after explosion. In the first few moves back to radiative equilibrium. Even with the use of these days the SN Ia envelope is optically thick and compact. Another big time steps, the radiative transfer for a light curve of 50 days point is, in the first few days the SNe Ia light curves are quite has to be solved about 10 000 times. For normal model atmo- faint, and there are almost no observations of this early time sphere calculation in PHOENIX with the temperature correction phase that have been obtained. Thus, for the model light curve procedure, the radiative transfer equation has to be solved about calculation, it is adequate to start the light curve calculation a 100 times, at most. Therefore, the computation of a whole SN few days after the explosion. The initial structure is given by Ia model light curve is very computationally expensive, because the result of the explosion model simulation. The results of the the solution of the radiative transfer equation has to be obtained explosion model give the expansion velocities, density structure too many times. It takes a huge amount of computation time to and the non-homogeneous abundances of all chemical elements calculate even simple SN Ia model light curves, even when using present in the SN Ia envelope. The envelope expands for the first only a few wavelength points. Basically, the SN Ia model light few days by assuming homologous expansion. The radii are de- curve is a curve consisting of half day points, where the atmo- termined by the expansion velocities and time after explosion. sphere is in radiative equilibrium. During the later phase after For the first temperature structure guess, the PHOENIX temper- maximum light, the big time steps have been performed for one ature correction procedure is used to obtain an initial tempera- or even two days, as the energy changes is the SN Ia atmosphere ture structure, which is in radiative equilibrium. Alternatively we become smaller. could calculate RE models starting at day one. As described above, a model spectrum is calculated with the For the computation of an SN Ia model light curve, we let obtained atmosphere structure, which is in radiative equilibrium the energy solver change the obtained initial model atmosphere at each half day point of the light curve. To obtain a model light structure. The atmosphere structure adapts to the new condi- curve for different photometric bands, filter functions were used tions caused by γ-ray deposition and other energy effects. After to calculate the luminosity in different bands. In Fig. 7,anSNIa a certain time, the atmosphere eventually reaches the radiative model spectrum and the filter functions for the U, B, V, R and I bands are shown. These filter functions are described in Hamuy 1 http://suspect.nhn.ou.edu/ et al. (1992). With these filter functions, SNe Ia model light

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Fig. 9. The energy change due to the additional energy input for several Fig. 8. The energy change due to the expansion for several days. days. curves can now be obtained for these ﬁve diﬀerent photometric bands.

4.2. Light curves of LTE models For the first calculations of theoretical light curves, the model atmosphere of the SN Ia is considered to be in LTE. For a first approach to obtain model light curves this is adequate. Another rea- son for the assumption of LTE is that model light curve calculations with an atmosphere treated in NLTE use more computation time. In order to obtain an SN Ia model light curve in reasonable computation time, a model atmosphere in LTE is a necessary assumption. For the model light curve calculations presented in this section the following parameters were chosen. The model atmosphere is divided into 128 layers. The radiative transfer is solved including atomic lines of the Kurucz atomic data line list. Fig. 10. The energy change due to the radiation for several days. The number of wavelength points used for the solution of the radiative transfer is about 2400. For the inner boundary condition of the radiation, we use the nebular boundary condition, which means that the radiation is just passing through the inner empty region, therefore all the radiation is produced by the ejecta itself, there is no inner lightbulb. The outer boundary condition is zero. For the first LTE light curve calculation, the W7 deflagration explosion model is used to obtain model light curves of SNe Ia in different photometric bands. The first point of the light curve was calculated at three days after the explosion. The method to obtain the theoretical light curves is described in Sect. 4.1. Before we present our resulting model light curves, we present plots of various terms that change the energy density for several moments in time. In Fig. 8, the energy change due to the expansion is shown for several moments in time. The energy change is decreasing for later days in the light evolution. The additional energy input due to the radioactive decay is shown in Fig. 9. With increasing time, the amount of energy input into the atmosphere decreases. It also shows, that the additional energy Fig. 11. Temperature profiles as a function of time. is located in the inner part of the envelope, where the 56Ni has been produced. Figure 10 shows the energy loss due to the emission of radiation. The temperature structures of the atmosphere to be later than that of the observed light curves. At 20 days after for several moments in time are shown in Fig. 11. the explosion, the model light curve has its maximum, while the Figure 12 shows the LTE SN Ia model light curve of the W7- maximum of the observed light curves is around 17 days after the based explosion model in the optical V band. The theoretical explosion. After maximum, the decline of the light curve of the light curve accurately reproduces the observed light curves of W7-based model well reproduces the observed light curve. Even two SNe Ia. The steep rise of the model light curve beginning at up to the later phase at 50 days after the explosion, where the three days after explosion is in agreement with the observed light atmosphere becomes significantly thinner, the fit to the observed curves. The maximum of the W7-based model light curve seems light curves is quite accurate.

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Fig. 12. LTE model light curve of the W7 explosion model in the V Fig. 14. LTE light curves of the W7 explosion model. The B band model band compared to two observed SN Ia light curves of SN 1999ee and light curve is too bright during the ﬁrst few days. SN 2002bo.

Fig. 15. The theoretical light curve in the R band seems to rise after a Fig. 13. U band light curve of the LTE model light curve compared to ﬁrst decline after the maximum phase. SN 1999ee.

The theoretical light curve in the ultraviolet U band is shown model light curve rises further, which is not seen in the observed in Fig. 13. Only an observed light curve of SN 1999ee is avail- light curves of SN 2002bo and 1999ee. Around day 30, the dif- able. The observational data are scattered. The rise in the begin- ference between model and observed light curve in the I band are ning as well as the maximum phase is well represented by the about 1 mag. Up to day 50, the model light curve declines, while model light curve. However, the decline of the theoretical light the observed light curves show their second maximum around curve seems to be too steep. This same effect is present in the 40 days after explosion. model light curve of the B band, which is shown in Fig. 14.The In Fig. 17, we compared the relation U − B of the theoretical first days of the model light curve are too bright compared to and observed light curves. The same comparision for the B − V both observed SN Ia light curves. The maximum phase of the relation is shown in Fig. 18. model light curve is in good agreement with the observed ones. At day 50, the model light curve becomes brighter than the ob- 4.3. Dynamical models served light curves. In Fig. 15, a plot of the model light curve of the R band is We have three different results of explosion calculations of SN shown. The steep rise in the beginning and the maximum phase Ia events. The structure of these models are results from hydro- of the observed SN Ia light curves is well represented by the dynamical explosion calculations. One is the W7 deflagration computed model light curve. However, the theoretical light curve model of Nomoto et al. (1984). The W7 dynamical model as- fit becomes worse for later phases. The luminosity of the model sumes that the ongoing explosion is a deflagration. The flame light curve seems to rise again at around day 25 after the explo- propagates with a velocity lower than the speed of sound out- sion. Up to day 45, the model light curve has a second bump, wards. Another possible explosion model is the delayed detona- which is not observed in the light curves of SN 1999ee and SN tion model. The explosion starts with a deflagration which even- 2002bo. In the infrared I band, the decline after the maximum tually proceeds to a detonation. We used two different dynamical phase is missing, as shown in Fig. 16.AsintheR band, the rise models named as DD25 and DD16. For more detailed informa- in the beginning and maximum are well represented in the model tion about the used delayed detonation models see Höflich et al. light curve. However, at maximum, the luminosity of the SN Ia (2002).

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Fig. 18. Comparison of B − V for the theoretical and observed light Fig. 16. LTE light curves of the W7 explosion model. For the I band curves. the model light curve has no distinctive maximum. At 30 days after the explosion the model light curve is way too bright in comparison to the observed ones.

Fig. 19. V band light curves for three diﬀerent dynamical models W7, DD16 and DD25. The DD25 dynamical model is fainter because less Fig. 17. Comparison of U − B for the theoretical and observed light nickel was produced during the explosion. The W7 and DD15 dynami- curves. cal model light curves are quite similar.

The resulting optical model light curves of the V band are light curves need improvements especially for the later phase plotted in Fig. 19. The light curve of the DD16 is fainter than the to fit the observational light curves more accurately. So far, the other ones. This is due to the fact that in this explosion model 56 SN Ia model atmosphere is considered to be in LTE during the less Ni is produced. The W7 and DD25 model are quite similar. whole evolution time. In this section, we calculate the model DD25 is slightly brighter than W7. The shape of the light curves light curves with the assumption of an atmosphere which is not are in good agreement with the observed ones. The maximum of in LTE. The computation of model light curves with SN Ia model the DD16 is later than the one of the W7 and DD25. atmospheres in NLTE requires a huge amount of computation In Fig. 20 the light curves in the I band of the three dynami- time. At first, we model light curves with atmospheres in NLTE cal models are shown. The infrared light curves deviate strongly that are computed with LTE temperature structures. More real- from the observations. As in the optical light curve, the DD16 istic NLTE model light curves with a temperature structure that light curve is fainter than W7 and DD25. DD16 also has only adapts to NLTE conditions are computed to investigate the NLTE one maximum and a decline. However, this maximum is later effects on the model light curves. than the observed ones. The DD25 and W7 light curves are still The first approach to compute NLTE model light curves is rising after the observed light curves have reached their maxi- to consider the atmosphere to be in NLTE, but use a fixed LTE mum and are declining. The theoretical light curves in the in- temperature structure. For this computation of NLTE model light frared must be improved for all explosion models. curves, we used the calculated radiative equilibrium LTE temperature structure of the W7 deflagration model. We keep this 5. Light curves of NLTE models temperature structure constant and perform 20 iterations to let the NLTE converge, which is mainly the occupation numbers The LTE model light curves in the V Band and most other of the species that are considered for the NLTE calculation. We bands are in quite good agreement with observed SN Ia light considered the following species for the calculations in NLTE: curves. However, in the near-infrared of the I band, the model H I, He I, He II, C I-III, O I-III, Ne I, Na I, Mg I-III, Si I-III,

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Fig. 20. I band light curves for the dynamical models W7, DD16 and Fig. 22. Model light curves of the W7 model in the U band. The NLTE DD25 are compared to the two observed SN Ia light curves. model atmosphere has an LTE temperature structure.

Fig. 21. Model light curves of the W7 explosion model in the V band. Fig. 23. Model light curves of the W7 model in the B band. The NLTE The NLTE model atmosphere has an LTE temperature structure. model atmosphere has an LTE temperature structure.

The NLTE model light curve of the U band is shown in S I-III, Ca II, Ti II, Fe I-III and Co II. These are the species that Fig. 22. The NLTE light curve shows almost no deviations from are most abundant in SN Ia atmospheres and mainly contribute the LTE light curve. During the later phase the steep decline to the spectrum. There is only little H and He in SN Ia, and in is also present in the NLTE model light curves. In Fig. 23,the W7, there is none. However, considering H I, He I and He II NLTE model light curve in the B band is presented. The NLTE in NLTE does not cost much computation time. The advantage model light curve is slightly fainter than the LTE light curve. The of this approach to an NLTE model light curve is that no tem- shape of the NLTE light curve seems to be the same as for the perature iterations have to be performed. This reduces the com- LTE light curve. The NLTE model light curve is also an accurate putation time significantly, although about 200 000 wavelength fit to the observed light curves. points are calculated instead of 2400 in case of LTE. In Fig. 24, the NLTE model light curve of the R band is In Fig. 21, the SN Ia model light curve in the V band of an shown. The luminosity of the NLTE light curve is fainter than the NLTE calculation with an LTE temperature structure is shown. LTE light curve. It has almost the same shape. But for the phase The NLTE light curve is fainter than the LTE light curve. During after 25 days after the explosion, the NLTE model light curve the maximum phase there is about 0.4 mag difference between rises again. The assumption of NLTE does not improve the fit both light curves. We have to point out that for our NLTE to the observed light curves of SN 1999ee and SN 2002bo. The models, the energy is not conserved because the radiation does infrared NLTE light curve for the I band is shown in Fig. 25. not thermalize within the envelope. After maximum the NLTE During the maximum phase, the NLTE light curve is fainter model light curve approaches the LTE light curve and agrees than the LTE light curve. A distinctive maximum is missing with the observed light curves as well as the LTE light curve. in the NLTE model light curve. For the phase between day 15 However, the NLTE model light curve in the V band does not and day 25, the NLTE model light curve fits the observed SN agree with the observed light curves very accurately. With the Ia light curves very accurately. Here, there is a significant im- assumption of an atmosphere in LTE, a better fit is obtained, al- provement compared to the light curve of the LTE model atmo- though NLTE is more accurate. sphere. But, after day 25 the NLTE model light curve starts to

A141, page 9 of 11 A&A 528, A141 (2011)

Fig. 24. NLTE and LTE model light curves in the R band of the W7 Fig. 26. NLTE and LTE model light curves of the W7 explosion model model. The NLTE model atmosphere has an LTE temperature structure. in the V band.

Fig. 25. NLTE and LTE model light curves of the W7 model. The NLTE Fig. 27. NLTE and LTE model light curves of the W7 explosion model model atmosphere has an LTE temperature structure. In the I band the in the I band. NLTE light curve is somewhat of an improvement.

the computation time significantly. Nevertheless, the computa- rise and becomes too bright. This is the same problem that al- tion of an NLTE light curve needs significantly more time than ready emerged in the LTE light curve. LTE. For this computation of NLTE model light curves, about 200 000 wavelength points need to be calculated, compared to about 2400 wavelength points for a pure LTE light curve calcula- 5.1. NLTE atmosphere structures tion. To obtain an NLTE model light curve in a reasonable time, We perform a more realistic NLTE calculation of the SN Ia we started the calculation at day 10 after the explosion and used model atmosphere evolution. The temperature structure is now the LTE temperature structure as the initial atmosphere structure. changing and adapting to the new conditions of NLTE. The cal- The main focus is to check if the infrared light curves during the culation of an NLTE model light curve takes a huge amount later phase can be further improved. of computation time. Considerably more wavelength points are In Fig. 26, the NLTE and LTE light curves of the W7 defla- needed for the calculation of the solution of the radiative trans- gration model in the V band are shown. The maximum phase is fer. For all the species considered in NLTE, the rate equations not well fitted by the NLTE model light curve. The LTE light have to be solved. Note that a time step in the NLTE calcula- curve fits the observed light curves better. At day 20 the NLTE tion is not a real time step. The rate equation changes the energy and LTE light curves are almost the same. Compared to the of the atmosphere, but this is not included in the energy solver. NLTE light curve obtained with the LTE temperature structure, However, it is adequate as the goal is to obtain a temperature there are only small differences to the NLTE calculation where structure, where the atmosphere is in radiative equilibrium. the temperature structure adapts to the NLTE conditions. In a first try to calculate a more realistic NLTE light curve, In Fig. 27,theI band model light curves of NLTE and LTE numerous species up to calcium are considered to be in NLTE. are shown. During the maximum phase, the NLTE light curve is These are the species H I, He I, He II, C I-III, O I-III, Ne I, fainter than the LTE light curve. Between day 15 and day 25 the Na I, Mg I-III, Si I-III, S I-III and Ca II. Higher species are NLTE model light curve fits the observed light curves quite ac- neglected because they have more levels, which would increase curately. Here, the use of NLTE improves the fit to the observed

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curves. Blinnikov et al. (2006) had the same deviations in their model light curves compared to observations. Their light curve in the I band is also too bright and further rising, even after the maximum in observed light curves. Kasen (2006)presenteda detailed study of SN Ia light curves in the near IR. His I band light curve fits the observations better. He also showed how variable and dependent on different parameters the light curves in the near infrared can be. Three different dynamical models were compared to observed SN Ia light curves. The delayed detonation model DD 16 is unlikely to be the best explosion model because the model light curves are too faint to fit the observed light curves due to the low 56Ni mass. The results of the W7 model are the best fit to the observed light curves. For the later phase of the light curves, especially in the infrared, we need the atmosphere to be considered in NLTE. We showed that NLTE model light curves computed with an LTE Fig. 28. Light curves in the I band. The light curve with variable scat- temperature structure leads to improvements in the model light tering is fainter than the LTE light curve. curves. For future work more investigation of the infrared model light curves and the NLTE needs to be performed. The focus has to lie on the influence of line scattering on the shape of the light curves. However, we obtained this improvement also with light curves. It would also be interesting to look in detail on the the NLTE model light curve calculated with the LTE temperature spectral evolution during the free expansion phase. Further im- structure. At day 25 the NLTE model light curve starts to rise provements may be achieved with a full 3D radiative transfer again. Although the consideration of NLTE improves the model and energy solver. light curve in the I band, the problem with a rise in brightness after the maximum remains. Acknowledgements. This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG) via the SFB 676, NSF grant AST-0707704, 5.2. Line scattering and US DOE Grant DE-FG02-07ER41517. This research used resources of the National Energy Research Scientific Computing Center (NERSC), which The fits to the infrared light curves need to be improved, as they is supported by the Office of Science of the US Department of Energy under are too bright during the phase of 20 to 50 days after the explo- Contract No. DE-AC02-05CH11231, and the Höchstleistungs Rechenzentrum sion in all models. We found that a change in the ratio of line Nord (HLRN). We thank all these institutions for generous allocations of com- scattering to absorption in the LTE case improves the fits to the putation time. I band light curve. The resulting light curve is shown in Fig. 28. The line scattering is here roughly inversely proportional to the References density and was for the figure “hand tuned” to produce a good match. As the density becomes lower due to the ongoing ex- Benetti, S., Meikle, P., Stehle, M., et al. 2004, MNRAS, 348, 261 pansion, the scattering becomes more important, as expected. A Blinnikov, S. I., Roepke, F. K., Sorokina, E. I., et al. 2006, A&A, 453, 229 Bongard, S., Baron, E., Smadja, G., Branch, D., & Hauschildt, P. 2006, ApJ, 647, more detailed investigation of the IR line scattering and its ef- 480 fects on SN Ia light curve modeling will be a topic of future Bronder, T. J., Hook, I. M., Astier, P., et al. 2008, A&A, 477, 717 work. To treat this effect more accurately, very detailed NLTE Chotard, N., et al. 2011, A&A, submitted model atoms are required that include the important infrared Foley, R. J., Filippenko, A. V., & Jha, S. W. 2008, ApJ, 686, 117 lines. In the atomic data we have presently available, these lines Goldhaber, G., Groom, D. E., Kim, A., et al. 2001, ApJ, 558, 359 Hachinger, S., Mazzali, P. 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